Hard Discs on the Hyperbolic Plane

TL;DR

This paper investigates a model of hard discs on the hyperbolic plane, exploring how negative curvature influences disorder, extending theoretical frameworks, and validating predictions with simulations near dense packings.

Contribution

It introduces a tractable model of disordered hard discs on hyperbolic space, extending free area theory and virial expansion to curved geometries.

Findings

Derived the equation of state for hard discs on hyperbolic plane.

Extended free area theory and virial expansion to curved space.

Validated theoretical predictions with simulation data near isostatic packing.

Abstract

We examine a simple hard disc fluid with no long range interactions on the two dimensional space of constant negative Gaussian curvature, the hyperbolic plane. This geometry provides a natural mechanism by which global crystalline order is frustrated, allowing us to construct a tractable model of disordered monodisperse hard discs. We extend free area theory and the virial expansion to this regime, deriving the equation of state for the system, and compare its predictions with simulation near an isostatic packing in the curved space.